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What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

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Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#21

Post by vishnugupta »

threatofrain wrote: post #3

The teacher mentions juggling units when adding and multiplying, but is 2 cm × 3 cm the same thing as 2 × 3?

> but is 2 cm × 3 cm the same thing as 2 × 3?

This made me think too, in particular the teacher's example. So; here's what I think it should be calculated.

3 cm + 3 cm = 2 X 3 cm (and not 3 cm + 3 cm = 2 cm X 3 cm).

What you are doing above is adding two objects each of who are 3 cm long. And the way you compute "2 X 3 cm" is "(2 X 3) cm" which then becomes "6cm".

However 3 cm + 3 cm != 2 cm X 3 cm as the author states which is what leads them to wrong conclusion. In other words 2 cm X 3 cm leads to an entirely new thing in geometry which is "area". So while 3 cm + 3 cm is still length (and hence it should be translated to 2 X 3 cm i.e., one scalar unitless number and other number with unit) 2 cm x 3 cm is not length. And the resultant area is correctly written as 6 cm^2; by multiplying units as well as the numbers. Now, how does one multiply numbers (2 X 3)? It's indeed by repeated addition!

The author, however is onto something. Which is when you multiply a something with unit (like 3Kg, 2Cm) with a unitless number that resultant product is still something (that weight, length) that you began with. However when you multiply two things both of which have units the resultant product is something entirely new (i.e., area as opposed to length, pressure as opposed to weight etc.,)

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#22

Post by bmc7505 »

wizzwizz4 wrote: post #12
bmc7505 wrote: post #10

Does anyone have an intuition for why Presburger arithmetic is decidable, but once you introduce x, it is not? If + can be rewritten as repeated succ, (which is decidable) and x can be rewritten as repeated +, then it seems logical that x should be decidable, but as any undergraduate CS student knows, Peano arithmetic is undecidable. What is so special about x exactly?

It's a variable number of addition operations.

Isn’t + a variable number of succ’s? Why does x break decidability, but + does not? It seems to me you should be able to define a second order Presburger system with + replacing succ, and x replacing + to restore decidability. Or is there some issue with countability or assumption that is broken?

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#23

Post by t0r0nat0r »

Garlef wrote: post #13

My favourite way of thinking about these operations comes from category theory: * addition: the cardinality of the disjoint union of two sets * multiplication: the cardinality of the direct product of two sets * exponentiation: a^b is the cardinality of the set of maps from B to A where B is a set of cardinality b and A is a set of cardinality a

This seems a bit restricted. How do you go about extending it to real and complex numbers?

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#25

Post by nicky0 »

Garlef wrote: post #18
Rerarom wrote: post #15

Earlier quoted context omitted.

This is just set theory, not category theory.

Neither is more foundational than the other. So I think the use of "just" is not justified here. https://mathoverflow.net/questions/360578/category-theory-an...

But talking about category theory makes you sound smarter.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#26

Post by salawat »

cturner wrote: post #6
threatofrain wrote: post #3

The teacher mentions juggling units when adding and multiplying, but is 2 cm × 3 cm the same thing as 2 × 3?

Both examples use the same operator, but we resolve them to different functions. Mentally, we use the tuple of argument types to select the appropriate function. Addition does not provide equivalent functions for all multiplication cases. Hence, multiplication is not repeated addition.

You need to stop conflating the tyoe mumbo jumbo with multiplication.

We don't define it based on what machines do with it, we define it based on what WE do with it. In the context of teaching, dimensional analysis comes far later in the process of education.

The pedants answer is that there is more than one multiplication operator taught. It's most basic incarnation is, for all intents and purposes, repeated addition.

It's more advanced forms, which include extra auditing for dimensional analysis isn't merely repeated addition. Though I'd challenge that too.

Dimensional analysis is an extra aspect stapled onto and seperate from the arithmetic operation. Therefore makes up an additional level of auditing seperately as a whole, and therefore inappropriate to define as an inherent property of the operator. You multiply numbers.

People need to stop trying to teach unrelated things bundled by some programmer or language writer to kids/adults. It doesn't make one seem smart. It just confuses people by blurring orthogonal aspects of problem solving, which puts people off of it.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#27

Post by rustybolt »

These articles bring back memories of teachers telling me I was wrong, because I didn't use the quadratic formula, but completed the square instead.

You can definitely see multiplication as repeated addition. It's even a useful thing to do, how else would you define multiplication?

This is the whole idea that made mathematics great. You define more complicated operations in terms of familiar operations. Then, if you gained enough intuition you are ready to use these more complicated operations as primitives as well. Then, you can generalize and apply your intuition to more abstract objects.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#28

Post by eigenket »

Garlef wrote: post #18
Rerarom wrote: post #15

Earlier quoted context omitted.

This is just set theory, not category theory.

Neither is more foundational than the other. So I think the use of "just" is not justified here. https://mathoverflow.net/questions/360578/category-theory-an...

Yeah but your comment didn't use the concept of a "category" at all, whereas you used "set" three times. Its definitely more natural to say these definitions come from set theory.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#29

Post by yakubin »

rustybolt wrote: post #27

These articles bring back memories of teachers telling me I was wrong, because I didn't use the quadratic formula, but completed the square instead. You can definitely see multiplication as repeated addition. It's even a useful thing to do, how else would you define multiplication? This is the whole idea that made mathematics great. You define more complicated operations in terms of familiar operations. Then, if you…

> You can definitely see multiplication as repeated addition.

Only for rational numbers. Doesn't work for real and complex numbers.

> It's even a useful thing to do, how else would you define multiplication?

Axiomatically, not algorithmically.

Re: What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

#30

Post by bmc7505 »

eigenket wrote: post #19
bmc7505 wrote: post #10

Does anyone have an intuition for why Presburger arithmetic is decidable, but once you introduce x, it is not? If + can be rewritten as repeated succ, (which is decidable) and x can be rewritten as repeated +, then it seems logical that x should be decidable, but as any undergraduate CS student knows, Peano arithmetic is undecidable. What is so special about x exactly?

Its probably worth pointing out that the system with only multiplication and no addition is also decidable (Skolem arithmetic). Things get interesting when you have both multiplication and addition defined and the reason for that is basically the fundamental theorem of arithmetic which gives you unique prime factorisation and directly the Gödel encoding (writing out statements about integers as integers). The fundame…

That’s a great explanation, thank you!
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